As the distance function of complexity geometry, the operator size measures how many local d.o.f. can be changed by an operator, but it does not measure the number of d.o.f. that are really changed. In general, acting on a state, only part of the operator matters. For example, \(\sigma ^x\otimes \sigma ^z|00\rangle =\sigma ^x\otimes I|00\rangle \) , where \(\sigma ^x\)  and \(\sigma ^z\)  are Pauli operators, \(I\)  is the identity. So the operator size should be modified with the knowledge of the state. With this observation, we define state-dependent operator size (SDOS) such that operators with the same action on a given state have the same size. Thus SDOS induces a metric on the Hilbert space with which we can define the complexity geometry of states. Unlike operator complexity geometry, it's not right-invariant and thus more difficult to calculate curvatures. But using Page's theorem [1]}, we can show that for \(N\) -qubit system, 2-surfaces generated by Hamiltonians of size much smaller than \(N/2\)  acting on typical states are negatively curved.
